Research on Assignment of Railway Passenger Train Set Based on Simulated Annealing Algorithm

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1 Journal of Information & Computational Science 10:15 (2013) October 10, 2013 Available at Research on Assignment of Railway Passenger Train Set Based on Simulated Annealing Algorithm Changfeng Zhu, Yifan Xiao School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou , China Abstract Optimization of railway passenger train set assignment is a complicated system engineering that is influenced by multitudinous factors. To improve the use efficiency of train set, the impact of train delay on train set assignment was analyzed, and the optimization model of railway passenger train set assignment have been built according to train delay propagation, and on that basis, optimization algorithm have also been put forward based on simulated annealing algorithm. Finally, a case study has been carried out taking four passenger s in railway network as an example in order to testify validity, objectivity and applicability of this model by using calculating and comparing analysis. The results show that this model could improve the efficiency of railway passenger train set assignment. Keywords: Passenger Train; Optimization Model; Algorithm Passenger train sets are the important resources in the railway transportation organization. Proper train sets assignment has significant meaning to lower railway operation cost. Recently, with development of railway transportation, the operating passenger train numbers are increasing. But, improper train sets assignment leads to long stop at passenger s, low operation efficiency of train sets and high operation cost of train sets. Dual-objective assignment model was established [1]. Integer programming model of which objective was maximum transportation capacity was proposed [2]. Optimal models were studied on uncertain railway region of highspeed trains [5-7]. However, most of the research was on the high-speed trains. Research on the ordinary passenger trains mostly focused on operation cost. Considering operation of different grades of train sets and passenger trains, train delay propagation etc, the objective model based on number of minimum passenger train set was set up. 1 Variables Definition and Problem Description m(m = 1, 2,, M) represents railway passenger ; r(r = 1, 2,, R) represents a train set; Project supported by The General Planning Project of Humanities and Social Sciences from Ministry of Education of China (No. 11YJAZH132, No. 11YJCZH170) Corresponding author. address: zhudd003@163.com (Changfeng Zhu) / Copyright 2013 Binary Information Press DOI: /jics

2 4894 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) V = {v i v i = t d i, t f i, tf i, l i, xi d i, ξi a } is trains set, where t d i, t f i is departure and of train i, respectively; t f i is the actual of train i; l i is the grade of train i, ξi d is the departure passenger of train i, ξi a is the passenger of train i; c k r represents that the grade of train set r is k; c m r represents that the standby train set r belongs to passenger m; µ i represents the train delay probability of train i; h i represents the related delay train number which are impacted by train i; T R represents standard technical operation of train sets in passenger s. Suppose there are m passenger s in the railway network. Ordinary passenger trains, express passenger trains and high-speed passenger trains can take over the passenger trains of which grade are not lower than their grades. However, the operation of standby train sets doesn t follow the constraint on grade of trains. During train s linkage processing, train delay problem should be considered. For a period of 24 hours, the train set assignment problem can be translated into the TSP problem with multi-constraints. 2 Established Optimal Model of Railway Train Set Assignment Suppose that train sets are classified into two categories, one category is unfixed train set and the other is standby train set which belongs to passenger s. And unfixed train sets can run among the whole railway network without constraints. And standby train sets should return to specified passenger s. Based on theory of train delay propagation and principle to simplify the problem, delay trains impact other train set operation when train delay occurs in the. And train delay occurs in other s, by default, trains can run on finally. (1) Regulation of railway schedule is that one passenger train can only be assigned with one train set and the grade of the passenger train cannot be lower than the linkage passenger train. Define decision variable x ij to judge if passenger train i take over passenger train j Then x ij = { 1, passenger train i joins passenger train j 0, or else Define l i represents the grade of train i 1, the grade of i is ordinary passenger train l i = 2, the grade of i is fast speed passenger train 3, the grade of i is exp ress passenger train Define C k r represents the grade of unfixed train set r 1, the grade of train set r is ordinary passenger train c k r = 2, the grade of train set r is fast speed passenger train 3, the grade of train set r is exp ress passenger train

3 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) According to related regulation, on train set can only takeover one passenger train and the grade of train set is higher than or equal to the grade of passenger train, then the expression can be denoted as N N x ij = 1 i=1 j=1 i, j, k (1) l i ck r 0 (2) Train sets are needed to have technical operation after their. The between two linkage trains should satisfy technical operation T R. The higher weighing value is, then the smaller linkage probability. At the same, the same grade passenger trains have the highest linkage probability. Define 0-1 variable P i p i = { 0, train iruns on 1, or else Then l [ j t d j t f i l p i ω ij = i l j [t d j t f i l p i i ( )] t f i t f i ) ( t f i t f i t d j max(t f i, tf i ) T R ] t d j max(t f i, tf i ) T R (2) (3) When the whole turnover of operation line that train i belongs to is longer than 24 hours, then the train set which take over the traini can only take over other trains in the next period. So, only train sets which complete operation in 24 hours can take over trains operation. 2t i + 2T R + t f i 1440 (3) (4) When passenger trains delays, due to the impact of train delay propagation, the number of related delay passenger trains can be denoted as Define 0-1decision variable s i s i = { 1, ξ a i = n 0, or else Then the number of standby train sets in n can be denoted as N k k n = max s i µ i h i (4) (5) Standby train sets are needed to return to specified passenger s when take over operation. When train i which is taken over by train set r take over train j, it must meet the following constraint i=1

4 4896 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) Define decision variable q r ij q r ij = { 1, train set r is standby 0, train set r is unfixed (c n r ξj a )(c n r ξj d ) = 0 N N x ij = 1 qij r i=1 j=1 (5) The objective of the model is: The total minimum stop at s can be denoted as min z m = Nk i=1 Nk (1 qij)l r i ω ij x ij /l j j=1 The total number of needed train sets N train-set can be denoted as N train-set Nk M N train-set = (t f i td i ) + z m / M Nk max i=1 m=1 m=1 i=1 k n i 3 Algorithm Design In this algorithm, matrix Y = (Y 1, Y 2, Y 3,..., Y n ) represents solutions. Expression Y i = j represents train I take over train j. and every passenger train use natural coding. Based on solution expression form, two mutual exchange rule was adopted. To one current solution, choose two trains of which have different s to exchange their operation lines and new solution can be generated. Choose one railway train schedule and number passenger trains. Initial solution is the number of departure train which corresponds to the train. For example, 1303 train (number i) will take over departure 1304 train (number Y i ) operation with a period of service. Based on expression f/t 0 0, f is increment of objective function value after neighborhood moving. Objective function in the paper is only related to and departure of two mutual linkage trains. According to analysis, with every neighborhood moving, the value of f is one of the following data, namely, 0, ±1440. Based on expression f/t 0 0, t 0 is higher the better. Applying T = T r cooling temperature mode, this cooling temperature mode has the characteristic that the decrease of temperature is fast with high temperature, and decrease of temperature is slow with low temperature. In the theory, ending temperature is the lower the better. The best temperature is 0 C. When the ending temperature is lower than C, then cooling process will end. Furthermore, neighborhood is generated by data exchange where the number of neighborhood moving is bigger than C 2 n. According to the problem in the paper, constrains processing cannot impact on algorithm performance and computational complexity. In the paper, refuse strategy is adapted to remove

5 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) the scheme which doesn t satisfy constraints hours is translated into min and the departure and of passenger trains also is translated into minutes. Step 1 specify name of data files, initial temperature, cooling temperature coefficient, ending temperature, heat balance s and other parameters. Step 2 read departure and of passenger trains from data files. Step 3 give an initial solution S 0 as current solution, T = T 0, c i = 1 Step 4 judge if passenger trains i, jare on. If trains are delay, then P i = P j =1. Or else P i = P j =0. Step 5 judge if two linkage passenger trains i, j are standby train sets. If they are standby trains, then p r ip i = p r jp j = 1. Or else, p r ip i = p r jp j = 0 Step 6 neighborhood solution is generated by exchanging Y i, Y j. New solution S is examined for if it satisfies constraints. The constrains in the paper are: (1) Regulation of railway schedule is that one passenger train can only be assigned with one train set and the grade of the passenger train cannot be lower than the linkage passenger train (2) Time between two linkage passenger trains should meet constraints. If above two requirements can be satisfied, then it turns to Step 7. Or else, it turns to Step 8. Step 7 calculated f which represents difference between f(s) and f(s 0 ). If f < 0, then S 0 = S. If f >0, random number x is generated in the interval (0, 1). If exp( f/t ) > x, then S 0 =S. If f(s 0 ) is the globally best solution, then update solution. Or else, it turns to Step 8. Step 8 ci = ci + 1. if c i < Q, it turns to Step 5. Or else, it turns to Step 9. Step 9 if T > , then it turns to Step 3. Or else, the whole computation process ends. 4 Empirical Analysis Take A, B, C, D four s in the railway network for example. The location of the s is illustrated in Fig. 1. The train routing of railway network is shown in Table 1. Information of passenger train flow of four s is shown from Table 2-5. A D 1876 km 881 km 670 km 571 km B C Fig. 1: Location of railway passenger s Based on information of train flow of four s, stop at s of train set can be obtained. The number of needed train sets is 36, where the number of ordinary train sets, express train sets and high-speed train sets is 6, 16, 12, respectively. Stop at s of railway passenger trains is shown in Table 6, and result of optimized train set assignment is shown in Table 7 and Table 8.

6 4898 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) Table 1: Train routing of railway network Train routing A-B A-B B-A B-A C-A C-B-A A-C A-B-C B-C B-C C-B C-B A-D A-D A-B-C B-D B-C-D C-D C-A Table 2: Information of passenger train flow of A passenger Train number Departure departure Train delay probability Delay trains impacted by precious trains 1303 A B / 20:35 / / 1481 A B / 21:23 / / K117 A B / 11: K179 A B / 22:37 / / T817 A B / 8:47 / / T167 A B 14:54 / / T201 A B / 18:09 / / K133 A C 21:51 / / T231 A C 6:29 / / K2061 A D / 14:36 / / T151 A D / 17:49 / / T175 A D / 12:08 / / 1304 B A 19:33 / B A 8:01 / K118 B A 5:36 / K180 B B 6:17 / T818 B A 22:52 / T168 B A 13:00 / T202 B A 6:35 / K134 C A 5:19 / K2062 D A 13:18 / T152 D A 15:31 / T176 D A 08:21 / By computation, stop at s of adjusted train set assignment is min which is less than 7602 min comparing with original train set assignment. The optimized train set assignment can save 5 operation train sets. However, there are 4 standby train sets in 4 s, respectively. So optimized train set assignment can totally save 1 train set.

7 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) Table 3: Information of passenger train flow of B passenger Train number Departure departure Train delay probability Delay trains impacted by precious trains 1304 B A / 7:20 / / 1482 B A / 22:55 / / K118 B A / 22:03 K180 B A / 22:12 / / K186 B A 00:23 / / K750 B A / 23:29 / / T818 B A / 14:00 / / T168 B A 05:36 / / T202 B A / 01:02 / / 1917 B C / 06:00 / / T197 B C / 21:26 / / T137 B C / 10:28 / / K125 B C / 13:17 / / K131 B D / 15:13 / / 1303 A B 07:39 / 0.3 3/ 1481 A B 07:49 / K117 A B 20:48 / K179 A B 08:07 / T817 A B 18:28 / T167 A B 21:54 / T201 A B 01:09 / C B 22:03 / T198 D B 06:06 / T138 C B 16:18 / K126 C B 18:40 / K132 D B 03:30 / Conclusion By studying on train delay propagation, passenger train set assignment model has been established in the paper. The model has the characteristic of simple computation and good operability. In reality, there exist balance usage of and departure lines and servicing lines which is the emphasis in the next stage.

8 4900 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) Table 4: Information of passenger train flow of C passenger Train number Departure departure Train delay probability Delay trains impacted by precious trains T232 C A / 18:30 / / T134 C A / 14:03 / / 1918 C B / 14:33 / / K126 C B / 12:30 / / K2062 D A 17:50 18:03 / / T151 B D 07:16 07:37 / / T152 B D 03:10 03:31 / / K131 B D 01:03 01:23 / / K132 D B 17:20 17:40 / / T197 B D 3:37 03:45 / / T198 D B 00:18 00:28 / / 1917 B C 22:03 / T231 A C 20:35 / T133 A C 13:51 / B C 13:30 / K125 B C 19:27 / Table 5: Information of passenger train flow of D passenger Train number Departure departure Train delay probability Delay trains impacted by precious trains K2062 D A / 08:50 / / T176 D A 10:29 14:21 / / K132 D B 18:10 18:24 / / K176 D B 08:00 08:14 / / T152 D B / 18:31 / / T198 D B / 16:06 / / K2061 A D 20:05 / T175 A D 06:20 / K131 B D 09:33 / T151 B D 14:49 / T197 B D 11:26 / 0.1 2

9 C. Zhu et al. / Journal of Information & Computational Science 10:15 (2013) Table 6: Stop at s of railway passenger trains Train number Trains operation interval Stop at A (min) Stop at B (min) Stop at C (min) Stop at D (min) 1303/1304 A B /1482 A B K117/K118 A B K179/K180 A B K817/K818 A B T167/T168 A B T201/T202 A B T133/T134 A C T231/T232 A C K2061/K2062 A D T151/T152 A D T175/T176 A D /1918 B C K131/K132 B D T197/T198 B D K125/K126 B C T137/T138 B C /1044 B C total Table 7: Optimized train set assignment Arrival train number 1304 T818 K134 K180 K118 T232 T168 T Linkage departure train number 1303 T231 K817 K117 K133 T167 K179 T Stop at s (min) Arrival train number K2062 T152 T176 T198 K126 T138 K132 Linkage departure train number K2061 T151 T175 T197 T818 T137 K131 Stop at s (min) Table 8: Optimized train set assignment Arrival train number 1303 T231 T817 K117 K133 T232 T179 T Linkage departure train number 1304 T138 K118 K180 T232 T167 T133 T Stop at s (min) Arrival train number K K2062 T151 T175 T197 K125 T137 K131 Linkage departure train number T K2061 T152 T176 T198 K126 T138 K132 Stop at s (min)

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